Theorem dnnumch2 40786

Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
dnnumch.f	⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dnnumch.g	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))

Assertion

Ref	Expression
dnnumch2	⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem dnnumch2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.f	. . 3 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2		dnnumch.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		dnnumch.g	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
4	1, 2, 3	dnnumch1 40785	. 2 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
5		f1ofo 6707	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → (𝐹 ↾ 𝑥):𝑥–onto→𝐴)
6		forn 6675	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
7	5, 6	syl 17	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
8		resss 5905	. . . . . 6 ⊢ (𝐹 ↾ 𝑥) ⊆ 𝐹
9		rnss 5837	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥) ⊆ 𝐹 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
10	8, 9	mp1i 13	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
11	7, 10	eqsstrrd 3956	. . . 4 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
13	12	rexlimdvw 3218	. 2 ⊢ (𝜑 → (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
14	4, 13	mpd 15	1 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530   ↦ cmpt 5153  ran crn 5581   ↾ cres 5582  Oncon0 6251  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  recscrecs 8172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173

This theorem is referenced by:  dnnumch3lem  40787  dnnumch3  40788